L(s) = 1 | − 2-s + 1.73·3-s + 4-s − 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s + 1.73·12-s − 13-s + 16-s − 17-s − 1.99·18-s − 1.73·22-s − 1.73·24-s + 25-s + 26-s + 1.73·27-s − 32-s + 2.99·33-s + 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s + 1.73·48-s − 50-s − 1.73·51-s − 52-s + ⋯ |
L(s) = 1 | − 2-s + 1.73·3-s + 4-s − 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s + 1.73·12-s − 13-s + 16-s − 17-s − 1.99·18-s − 1.73·22-s − 1.73·24-s + 25-s + 26-s + 1.73·27-s − 32-s + 2.99·33-s + 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s + 1.73·48-s − 50-s − 1.73·51-s − 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.613395911\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613395911\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.902171554130202618911694926166, −8.299576845134420884616515967567, −7.45507297238138286722127694965, −6.92857563484196563808035938517, −6.27744887887884551330707218521, −4.73206168258376016760797597991, −3.84131042076333239670533691248, −2.99710921713900790186056301790, −2.21260851307020926107644850563, −1.36709598565501862331938183107,
1.36709598565501862331938183107, 2.21260851307020926107644850563, 2.99710921713900790186056301790, 3.84131042076333239670533691248, 4.73206168258376016760797597991, 6.27744887887884551330707218521, 6.92857563484196563808035938517, 7.45507297238138286722127694965, 8.299576845134420884616515967567, 8.902171554130202618911694926166